Received: 30 August 2017 Revised: 24 May 2018 Accepted: 24 May 2018 DOI: 10.1002/htj.21337 RESEARCH ARTICLE Magnetic field and viscous dissipation effect on bioconvection in a permeable sphere embedded in a porous medium with a nanofluid containing gyrotactic micro-organisms Sachin Shaw 1 Sandile S. Motsa 2 Precious Sibanda 3 1 Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Palapye, Botswana 2 Department of Mathematics, University of Swaziland, Kwaluseni, Swaziland 3 School of Mathematics, Statistics and Com- puter Science, University of KwaZulu-Natal, Pietermaritzburg, South Africa Correspondence Sachin Shaw, Department of Mathematics and Statistical Sciences, Botswana International University of Science and Technology, Private Bag 16, Palapye, Botswana. Email: shaws@biust.ac.bw, sachinshaw@gmail.com Abstract Micro-organisms play a vital role to understand the ecolog- ical system and therefore it is very important to understand the behavior of micro-organism due to different parame- ters. In the present paper, we investigated the bioconvection about a permeable sphere with constant surface temperature embedded in a porous medium filled with a water-based nanofluid containing gyrotactic micro-organisms. The con- vection and movement of the micro-organisms are con- strained or assisted by thermophoresis, viscous dissipa- tion effects, and an applied magnetic field. First, we have used the similarity transformation to simplify the govern- ing equations. Then, we have solved the governing non- linear partial differential equations numerically using a recent spectral relaxation method. The effects of the sig- nificant parameters on the local density of the gyrotactic micro-organisms have been determined and discussed. It is observed that by introducing the magnetic field in the sys- tem, the skin friction, local nanoparticle Sherwood number, and the local density of the micro-organism decrease, while the last two governing parameters show a positive response with increase in the viscous dissipation. KEYWORDS bioconvection, gyrotactic micro-organism, magnetic field, nanofluid, per- meable sphere, viscous dissipation 718 © 2018 Wiley Periodicals, Inc. wileyonlinelibrary.com/journal/htj Heat Transfer—Asian Res. 2018;47:718–734.